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There is a whole number n for which the following holds: if you put a 4 at the end of n, and multiply the number you get in that way by 4, the result is equal to the number you get if you put a 4 in front of n. In other words, we are looking for the number you can put on the dots in the following equation: 4... = 4 × ...4 Which number must be put on the dots to get a correct equation?

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The number 101694915254237288135593220338983050847457627118644067796.

The fraction EVE/DID = 0,TALKTALKTALKTALK... is a normal fraction that can also be written as a recurring decimal. Which fraction is this (equal letters are equal ciphers)?

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The two solutions are:212/606=0,34983498.242/303=0,79867986.

A cable, 16 meters in length, hangs between two pillars that are both 15 meters high. The ends of the cable are attached to the tops of the pillars. At its lowest point, the cable hangs 7 meters above the ground. How far are the two pillars apart?

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Note that it is a kind of trick question: the pillars stand next to each other. Which means that the cable goes 8 meters straight down and 8 meters straight up. Conclusion: The distance between the pillars is zero meters.

The legendary king Midas possessed a huge amount of gold. He hid this treasure carefully: in a building consisting of a number of rooms. In each room there were a number of boxes; this number was equal to the number of rooms in the building. Each box contained a number of golden coins that equaled the number of boxes per room. When the king died, one box was given to the royal barber. The remainder of the coins had to be divided fairly between his six sons. Is a fair division possible in all situations?

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The legendary king Midas possessed a huge amount of gold. He hid this treasure carefully: in a building consisting of a number of rooms. In each room there were a number of boxes; this number was equal to the number of rooms in the building. Each box contained a number of golden coins that equaled the number of boxes per room. When the king died, one box was given to the royal barber.

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For the six brothers, N3 - N coins remain. We can write this as: N(N2 - l), or: N(N - 1)(N + l). This last expression is divisible by 6 in all cases, since a number is divisible by 6 when it is both divisible by 3 and even.This is indeed the case here: whatever N may be, the expression N(N - 1)(N + l) always contains three successive numbers.

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One of those is always divisible by 3, and at least one of the others is even. This even holds when N=1; in that case all the brothers get nothing, which is also a fair division! .

This is a most unusual paragraph. How quickly can you find out what is so unusual about it? It looks so ordinary that you would think that nothing is wrong with it at all, and, in fact, nothing is. But it is unusual. Why? If you study it and think about it, you may find out, but I am not going to assist you in any way. You must do it without any hints or coaching. No doubt, if you work at it for a bit, it will dawn on you. Who knows? Go to work and try your skill. Good luck! What is unusual about the above paragraph?

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The paragraph doesn't contain a single letter "e".

Here is a sequence of numbers: 1 11 21 1211 111221 It seems to be a strange sequence, but yet there is a system behind it... What is the next term in this sequence?

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Again, the system behind the sequence is that each number "describes" the previous number. Now, however, the number of occurrences of each cipher is counted. So 1231 means one "2" and three times a "1", and 131221 means one "3", one "2", and two times a "1". The number following on 131221 is therefore 132231 (one "3", two times a "2", and three times a "1").

Below are a number of statements: 1. Precisely one of these statements is untrue. 2. Precisely two of these statements are untrue. 3. Precisely three of these statements are untrue. 4. Precisely four of these statements are untrue. 5. Precisely five of these statements are untrue. 6. Precisely six of these statements are untrue. 7. Precisely seven of these statements are untrue. 8. Precisely eight of these statements are untrue. 9. Precisely nine of these statements are untrue. 10. Precisely ten of these statements are untrue. Which of these statements is true?

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The ten statements all contradict each other. So there can be at most one statement true. Now suppose there is no statement true. That would mean that statement 10 indeed would be true, which results in a contradiction. This means that exactly nine statements must be untrue, and thus only statement 9 is true.

Three salesmen went into a hotel to rent a room. The manager stated that he had only one room left, but all three could use it for $30.00 for the night. The three salesmen gave him $10.00 each and went up to their room. Later, the manager decided that he had charged the salesmen too much so he called the bellhop over, gave him five one-dollar bills, and said: 'Take this $5.00 up to the salesmen and tell them I had charged them too much for the room'. On the way up, the bellhop knew that he could not divide the five one-dollar bills equally so he put two of the one-dollar bills in his pocket and returned one one-dollar bill to each of the salesmen. This means that each salesman paid $9.00 for the room. The bellhop kept $2.00. Three times nine is 27 plus two is 29....... What happened to the extra dollar?

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The calculation just makes no sense. The three salesman paid $27, of which the manager got $25 and the bellhop $2. Conclusion: There's no dollar missing at all.

A man decides to buy a nice horse. He pays $60 for it, and he is very content with the strong animal. After a year, the value of the horse has increased to $70 and he decides to sell the horse. But already a few days later he regrets his decision to sell the beautiful horse, and he buys it again. Unfortunately he has to pay $80 to get it back, so he loses $10. After another year of owning the horse, he finally decides to sell the horse for $90. What is the overall profit the man makes?

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Consider the trade-story as if it describes two separate trades, where: In the first trade, the man buys something for $60 and sells it again for $70, so he makes a profit of $10. In the second trade, the man buys something for $80 and sells it again for $90, so he makes again a profit of $10. Conclusion: The man makes an overall profit of $10 + $10 = $20.

Joyce has bought ten trees for her garden. She wants to plant these trees in five rows, with four trees in each row. The Question :How must Joyce plant the trees?